Lagrange Undetermined Multipliers 1

7 Lagrange Undetermined Multipliers

This chapter is a mathematical interlude. It introduces a powerful method for solving maximum and minimum problems that have various auxiliary conditions. For example, we may analyze the maximum information that can flow through a wire subject to the condition that there is some average duration of symbol transmission. This method of Lagrange undetermined multipliers is widely applied throughout physics and engineering.

Maximum-Minimum Problems

We often want to know the maximum or minimum values of a particular function. Some examples; maximizing the range of a missile or minimizing the energy expended by a cruise ship. The basis for doing such problems is the fact that the slope of a function is zero at local maximum and minimum points (see the figure).

The condition for either the maximum or minimum of the function y(x) is

dydx

y x0 at max and min pts. of . (1)

1 Find the value of x for which A x x 8 2 is a maximum. Calculate the maximum value of A. [ans. x=4, Amax=16]

2 A rectangular area is to be enclosed by a sixteen meter fence. Find the dimensions of the fence that will enclose the greatest area. [ans. 4 m square]

3 A rectangular area is to be enclosed by a sixteen meter fence where one side of the rectangle does not need fence because it is adjacent to a river. Find the dimensions of the fence that will enclose the greatest area. [ans. 4m X 8m]

maximum

minimum

r i v e r

Lagrange Undetermined Multipliers 2

Basic Idea of Undetermined Multipliers

As an example, assume we are given a function F x y x y, that we are to maximize subject to the constraint f x y x y( , ) . 2 16 0 If F x y( , ) were not constrained (that is, if x and y were independent), we could insist that differential dF=0.

dF x y Fx

dx Fy

dy( , )

0 (2)

When x and y are independent, the coefficients of differentials must each vanish,

Fx

Fy

0 0 and (3)

These conditions are easy to work with. Alas, only two equations can be solved for the two variables x and y whereas there are three equations to be satisfied; Eqs.(3) and the constraint equation .

The idea behind undetermined multipliers is to introduce another quantity whose value we can choose to make x and y appear independent. Thus we introduce a third unknown in our three-equation example. In particular, since f x y( , ) 0 , maximizing the function

F x y F x y f x y, , ,

is equivalent to the original problem. However, now we have a new parameter essentially a third unknown that can be determined along with x and y to satisfy the three relations:

Fx

Fy

0 0 and

andf x y( , ) 0

exampleRepeat problem 3 using the method of undetermined multipliers. Label the sides as in the diagram and use the fact that the perimeter must equal 16 as the constraint equation.

solution area

constraintConstruct and require the following:

.

x

y

river

Lagrange Undetermined Multipliers 3

The latter equations together with the constraint are three equations that allow us to solve for the three unknowns, x, y, and . [ans. = –4]

4 Use the method of Lagrange multipliers to determine the dimensions of a box with a square base and without a top such that the total surface area is 12 and the volume is a maximum. [ans. base =2, height =1]

5. A cylindrical soup can has a total surface area A (given). Use the method of Lagrange multipliers to determine the radius r and height h corresponding to the maximum volume. [ans.

]

The General Case

The general problem is to maximize or minimize a function of N variables F x x xN( , , , )1 2 subject to a set of K constraint equations,

f x x xf x x x

f x x x

N

N

K N

1 1 2

2 1 2

1 2

00

0

( , , , )( , , , )

( , , , )

(4)

The method then is to introduce K undetermined multipliers, 1 2, , , K and form the quantity

F x x x F x x x f x x xN N ii

K

i N( , , , ) ( , , , ) ( , , , )1 2 1 21

1 2 (5)

Now treat all the variables as independent so that dF’ = 0 gives

FxFx

FxK

1

2

0

0

0

(6)

Equations (4) and (6) are N+K equations that we can solve to find x x xN1 2, , , and 1 2, , , K .

Lagrange Undetermined Multipliers 4

6 Find the values of x and y that give an extremum for the function

subject to the constraints

where the E’s and U are constants. Do not bother to evaluate the Lagrange multipliers. You can redefine the multipliers to make your results look neater; that is, you can combine them with other constants or multiply them by arbitrary constant factors.

answer:

where and is a Lagrange multiplier.

This problem is a prototype for the information approach to finding canonical probabilities introduced in the next chapter. It is important for you to be able to perform the required manipulations. The next page gives a detailed solution to this problem (as it was done in Maple).

Lagrange Undetermined Multipliers 5

[1]

[2]

Now require derivatives with respect to x, y, and z to equal zero:

Now solve the x-equation,

Let x be rewritten in the conventional form by redefining the Lagrange multipliers:Let and Quantities y and z will have similar solutions.Finally, substitute the solutions for x, y, and z into constraint [2] to find Z in terms of β ( is the only remaining undetermined parameter):